View source for Multiplicative semi-group

''of an associative ring''

The [[semi-group]] formed by the elements of the given associative ring relative to multiplication.  In a unital ring (ring with multiplicative identity) this is a [[monoid]].  A non-associative ring is, relative to multiplication, only a [[magma]]; it is called the multiplicative system of the ring.

Some properties of a ring can be expressed in terms of the multiplicative semigroup.  For example, a [[factorial ring]] is one with a multiplicative [[Gauss semi-group]].

''of a general magma''

If $M$ is a [[magma]] with a [[binary operation]] written $\cdot$ then the left and right multiplication operators are the maps from $M$ to itself defined by $L_x :y \mapsto x \cdot y$ and $R_x : y \mapsto y \cdot x$.  The multiplicative or multiplication semigroup of $(M,{\cdot})$ is the semigroup generated by the operators $L_x$ and $R_x$ within the semigroup of all maps from $M$ to itself.

Properties of the magma $M$ are reflected in those of its multiplication semigroup.  For example, the [[flexible identity]] is equivalent to each pair $L_x$, $R_x$ commuting, and [[associativity]] to all $L_x$ commuting with all $R_y$; $M$ is a [[quasi-group]] if each of the left and right multiplication operators are permutations.  

====References====
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  R.H. Bruck,   "A survey of binary systems" Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. '''20''' Springer  (1958) {{ZBL|0081.01704}}</TD></TR>
</table>

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